GREEN UNIVERSITY OF BANGLADESH

Department: EEE (Eve)

Prepared For:

Md. Joynal AbedinLecturer, GUB

Prepared By:

Shahida AkterID# 143010003

Course Title: Control System

Root Locus

DefinitionIn control theory and stability theory, 

Root Locus analysis is a graphical method for examining how the roots of a system change with variation of a certain system parameter, commonly a gain within a feedback system.

The root locus is the path of the roots of the characteristic equation traced out in the s-plane as a system parameter (K) is changed.(0<K<∞) It can be used to describe qualitatively the performance of a system as various parameters are changed It gives graphic representation of a system’s transient response and also stability We can see the range of stability, instability, and the conditions that cause a system to break into oscillation

What is Root Locus

Root Locus Concept…T(s) = 1 + K G(s) H(s)

K G(s)

Character eqn,

1 + K G(s) H(s) = 0K G(s) H(s) = -1= 1 180̊= 1 (2K + 1) 180̊

<<

Pole Angle: θ1 = Given Point – PoleZero Angle: θ2 = Given Point – Zero

Root Locus Concept…

Angle Contribution = Pole Angle – Zero Angle [if Angle Contribution is odd multiplication of 180̊, Point lies on Root Locus; else not]

Π Pole LengthΠ Zero Length

Corresponding Gain: K =

Defining the Root Locus

Test whether point P = -5 + j5 lies on root locus and also calculate gain.

R(s) C(s)s (s + 10) K

+-

T(s) = s + 10s + K

K

2 P = -5 + j5

б-10 0

θ1

θ2

θ1 = (-5 + j5) – 0 = 5√2 135̊θ2 = (-5 + j5) – (-10) = 5√2 45̊

<<

jω

Given point P = -5 + j5

Angle Contribution = θ1 + θ2 = 135̊ + 45̊ = 180̊Since, Angle Contribution is odd multiple of 180̊, point P lies on Root Locus.

Corresponding Gain: K =

= (5√2 + 5√2) / 1 = 50

Π Pole Length

Π Zero Length

R(s) C(s)s (s + 10)

K +

-